[Paper Review] Classification and threshold dynamics of stochastic reaction networks

Carsten Wiuf, Chuang Xu|arXiv (Cornell University)|Dec 14, 2020

Gene Regulatory Network Analysis28 references4 citations

TL;DR

This paper provides a complete classification and threshold analysis of one-dimensional stochastic reaction networks (SRNs) modeled as continuous-time Markov chains (CTMCs). It establishes sharp, checkable criteria—based on four key parameters—for explosivity, recurrence, ergodicity, and tail behavior of stationary or quasi-stationary distributions, proving that all weakly reversible, one-dimensional mass-action SRNs are positive recurrent with CMP-like stationary distributions, thus confirming the positive recurrence conjecture in dimension one.

ABSTRACT

Stochastic reaction networks (SRNs) provide models of many real-world networks. Examples include networks in epidemiology, pharmacology, genetics, ecology, chemistry, and social sciences. Here, we model stochastic reaction networks by continuous time Markov chains (CTMCs) and pay special attention to one-dimensional mass-action SRNs (1-d stoichiometric subspace). We classify all states of the underlying CTMC of 1-d SRNs. In terms of (up to) four parameters, we provide sharp checkable criteria for various dynamical properties (including explosivity, recurrence, ergodicity, and the tail asymptotics of stationary or quasi-stationary distributions) of SRNs in the sense of their underlying CTMCs. As a result, we prove that all 1-d endotactic networks are non-explosive, and positive recurrent with an ergodic stationary distribution with Conley-Maxwell-Poisson (CMP)-like tail, provided the state space of the associated CTMCs consists of closed communicating classes. In particular, we prove the recently proposed positive recurrence conjecture in one dimension: Weakly reversible mass-action SRNs with 1-d stoichiometric subspaces are positive recurrent. The proofs of the main results rely on our recent work on CTMCs with polynomial transition rate functions.

Motivation & Objective

• To classify all communicating classes in one-dimensional stochastic reaction networks (SRNs) with mass-action kinetics.
• To derive sharp, checkable criteria for dynamical properties such as explosivity, recurrence, and ergodicity in 1D SRNs.
• To resolve the positive recurrence conjecture for weakly reversible mass-action SRNs in one dimension.
• To characterize the tail asymptotics of stationary and quasi-stationary distributions in 1D SRNs.
• To establish that all endotactic 1D SRNs are non-explosive and positive recurrent with CMP-like tails.

Proposed method

• Models 1D SRNs as continuous-time Markov chains (CTMCs) with polynomial transition rate functions.
• Introduces four key parameters—R, α, β, γ—derived from reaction stoichiometries and rate constants to characterize system dynamics.
• Applies recent results on CTMCs with polynomial rates to derive necessary and sufficient conditions for non-explosivity and positive recurrence.
• Uses geometric and algebraic techniques to classify communicating classes and analyze the structure of the state space.
• Employs the Conley-Maxwell-Poisson (CMP) distribution as a reference for tail behavior, linking it to stationary and quasi-stationary distributions.
• Applies the sweep test and face analysis to verify endotacticity and weak reversibility in reaction networks.

Experimental results

Research questions

• RQ1 What are the complete communicating class structures in one-dimensional stochastic reaction networks with mass-action kinetics?
• RQ2 Under what parameter conditions is a 1D SRN non-explosive, recurrent, or ergodic?
• RQ3 Do all weakly reversible, one-dimensional mass-action SRNs exhibit positive recurrence with an ergodic stationary distribution?
• RQ4 What is the tail behavior of stationary and quasi-stationary distributions in 1D SRNs, and how does it relate to the CMP distribution?
• RQ5 Are all endotactic 1D SRNs non-explosive and positive recurrent?

Key findings

• All one-dimensional weakly reversible mass-action SRNs are positive recurrent, confirming the positive recurrence conjecture in this dimension.
• All one-dimensional endotactic SRNs are non-explosive and admit a unique ergodic stationary distribution with CMP-like tail decay.
• The stationary and quasi-stationary distributions of 1D SRNs exhibit a trichotomy in tail behavior: super-exponential (CMP-like), exponential (geometric), or sub-exponential (power-law).
• A necessary and sufficient condition for non-explosivity is derived in terms of four parameters, providing a sharp threshold criterion.
• For 1D SRNs with R=1 and α=0, the system reduces to a single state {0} with no positive reactions, leading to T={0} and P=∅.
• The stationary distribution of a 1D SRN has a tail that decays like a CMP distribution if and only if the system is essential and endotactic, with the tail behavior fully determined by the four parameters.